I think I finally understand what Hersh is driving at. I had seen all along
that he is obviously right about the social nature of mathematics, and had felt
that where he failed was in explaining the unique objectivity of mathematics
among all other areas of human thought. Now I see (and I seem to recall that
this was implicit in his earliest postings on the subject, but obscured by a
couple of weeks of arguments in which he and his critics misunderstood each
other) that in fact this is his DEFINITION of mathematics! So the real problem
is that this does not explain what's special about mathematics, it just posits
it. Therefore, in order to assail Hersh, we must have ready a definition of
mathematics that is independent of Hersh's, and show that it somehow works
better than his. But that's not enough, if it still turns out that his
definition is "extensionally" the same in that it ends up calling the same
things "mathematics" that ours does--then he merely needs to explain why there
should be any uniquely objective area of human thought. To refute him we must
find some math which his definition misses or non-math it includes!-Joe Shipman